Final Project
Mark Berman, Joel Kopp, and Richard Wheeler
7/23/2018
- Introduction
- The Search for Better Multiple Linear Regression Models For Predicting Season Wins
- Methodology
- Better Candidate Multiple Linear Regression Models
- Evaluation and Refinement of the Candidate Models
- Evaluation of the Backwards Search: BIC Model
- Refining the Backwards Search: BIC Model Due to High Variance Inflation Factors
- Evaluation of the Backwards Search: AIC Model
- Refining the Backwards Search: AIC Model Due to High Variance Inflation Factors
- Evaluation of the Step Search: BIC Model
- Refining the Step Search: BIC Model Due to High Variance Inflation Factors
- Evaluation of the Step Search: AIC Model
- Refining the Step Search: AIC Model Due to High Variance Inflation Factors
- Evaluation of the Forward Search: BIC Model
- Refining the Forward Search: BIC Model Due to High Variance Inflation Factors
- Evaluation of the Forward Search: AIC Model
- Refining the Forward Search: AIC Model Due to High Variance Inflation Factors
- Validate Model Effectiveness Using Test Data
- Results
- Discussion
- Methodology
- Logistic Modeling and Prediction of Division Winners
- Apendices
library(readr)
library(lmtest)
library(faraway)Introduction
Conventional wisdom is that money buys happiness (winning) in Major League Baseball. However, the advent of “Moneyball” in the early 2000s by the Oakland Athletics, Cleveland Indians, and other teams, has lead to a more analytical approach to determining the make-up of Major League rosters.
As it turns out, money is not the magic elixir when it comes to assembling a winning Major League Baseball (MLB) team. The following plot shows that salary does not highly correlate with a winning record. This is substantiated by the companion single linear regression model and summary statistics that show that salary, while significant, only has a marginal impact on wins by an MLB team. The adjusted \(R^2\) from the simple linear regression – using training data from 2000 through 2013 – is low. Furthermore, the average percent error that compares actual wins versus predicted wins from the the test data (2014 through 2016) is high.
##
## Call:
## lm(formula = W ~ salary, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -34.389 -8.180 0.701 8.404 35.659
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 71.70962155199 1.29065984697 55.560 < 2e-16 ***
## salary 0.00000011551 0.00000001468 7.866 0.0000000000000316 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.86 on 418 degrees of freedom
## Multiple R-squared: 0.1289, Adjusted R-squared: 0.1269
## F-statistic: 61.87 on 1 and 418 DF, p-value: 0.00000000000003158## [1] "Leave One Out Cross-Validated RMSE: 10.88"
So what are the factors that move the needle?
We explore two related threads in attempting to identify the factors that have a strong impact on a winning baseball team.
The first thread uses multiple linear regression to identify the factors that influence a team’s winning record over the course of a season. The second thread uses logistic regression to identify the factors that influence a team’s ability to win their division.
Data
A few words are in order about where we obtained the data from to perform this analysis.
The source of data is the Sean Lahman baseball archive (http://www.seanlahman.com/baseball-archive/), recognized by the Society for American Baseball Research (SABR) as the leading detailed player and team data archive from 1874 through the end of the 2017 Major League Baseball season. We use team statistics from the years 2000 through 2013 as the training dataset and use data from 2014 through 2016 as the test dataset. (As a note, we were unable to include the 2017 season in our analysis due to unavailability of salary data.)
Variables
The team-based variables we examine fall into one of three categories: offensive, defensive, or administrative.
Administrative
yearID- YearfranchID- Franchise, or team nameW- Wins (multiple linear regression response variable)DivWin- Division winner (Y or N factor- Logistic linear regression response variable)WCWin- Wild Card winner (Y or N factor)LgWin- League champion (Y or N factor)WSWin- World Series winner (Y or N factor)salary- Team Salary (U.S dollars not adjusted for inflation)
Offensive
R- Runs scoredAB- At batsH- Total hits (including doubles, triples, and home runs)X1B- SinglesX2B- DoublesX3B- TriplesHR- Home runsBB- Base on balls (walks)SO- StrikeoutsSB- Stolen basesCS- Caught stealingHBP- Hit by pitchSF- Sacrifice fliespark- Name of team’s home ballpark (factor)attendance- Home attendance totalBPF- Three-year park factor for battersGIDP- Grounded into double playsRBI- Runs Batted InIBB- Intentional walksTB- Total basesSLG- Slugging percentageOBP- On-base percentageOPS- On-base plus slugging percentageBABIP- Batting average for balls in playRC- Runs createduBB- Unintentional walkswOBA- Weighted on-base average
Defensive
RA- Total runs allowedER- Earned runs allowedERA- Earned run averageCG- Complete gamesSHO- ShutoutsSV- SavesIPOuts- Outs pitchedHA- Hits allowedHRA- Home runs allowedBBA- Walks allowedSOA- Strikeouts by pitchersE- ErrorsDP- Double playsFP- Fielding percentagePPF- Three-year park factor for pitchersWHIP- Walks and hits per innings pitched
The Search for Better Multiple Linear Regression Models For Predicting Season Wins
Methodology
Better Candidate Multiple Linear Regression Models
scope <- W ~ R + AB + H + X2B + X3B + HR + BB + SO + SB + CS + HBP + SF + RA +
ER + ERA + CG + SHO + SV + IPouts + HA + HRA + BBA + SOA + E + DP + FP +
attendance + BPF + PPF + salary + RBI + GIDP + IBB + TB + SLG + OBP + OPS +
WHIP + BABIP + RC + X1B + uBB + wOBA
formula <- formula(scope)
start_model <- lm(formula, data= bbproj_trn)
n <- length(resid(start_model))
step_search_start_model <- lm(W ~ 1, data= bbproj_trn)Backwards Search: BIC Model
### Backwards Search: BIC Model
bic_model <- step(start_model, direction = "backward", k = log(n), trace = 0)
summary(bic_model)##
## Call:
## lm(formula = W ~ R + AB + H + X2B + BB + SO + CS + SF + RA +
## CG + SHO + SV + IPouts + HRA + BBA + E + FP + GIDP + BABIP +
## RC, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.8573 -1.9757 -0.1555 1.9118 7.4927
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1297.971349 339.322389 3.825 0.000152 ***
## R 0.076579 0.006140 12.472 < 2e-16 ***
## AB -0.116786 0.014360 -8.133 5.37e-15 ***
## H 0.306262 0.049740 6.157 1.81e-09 ***
## X2B 0.034285 0.010085 3.399 0.000743 ***
## BB 0.031072 0.006669 4.659 4.33e-06 ***
## SO 0.051706 0.009680 5.341 1.55e-07 ***
## CS -0.043025 0.015571 -2.763 0.005990 **
## SF -0.095773 0.021708 -4.412 1.32e-05 ***
## RA -0.054254 0.003811 -14.238 < 2e-16 ***
## CG 0.148895 0.046962 3.171 0.001639 **
## SHO 0.150078 0.049080 3.058 0.002380 **
## SV 0.345616 0.025203 13.714 < 2e-16 ***
## IPouts 0.057453 0.007126 8.063 8.78e-15 ***
## HRA -0.026131 0.008530 -3.063 0.002338 **
## BBA -0.008042 0.002619 -3.070 0.002287 **
## E -0.182853 0.055690 -3.283 0.001116 **
## FP -1045.006972 343.178691 -3.045 0.002480 **
## GIDP -0.044072 0.012147 -3.628 0.000322 ***
## BABIP -760.159504 137.620611 -5.524 6.00e-08 ***
## RC -0.115957 0.024057 -4.820 2.04e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.729 on 399 degrees of freedom
## Multiple R-squared: 0.9475, Adjusted R-squared: 0.9449
## F-statistic: 360.1 on 20 and 399 DF, p-value: < 2.2e-16Backwards Search: AIC Model
### Backwards Search: AIC Model
aic_model <- step(start_model, direction = "backward", trace = 0)
summary(aic_model)##
## Call:
## lm(formula = W ~ R + AB + H + X2B + BB + SO + CS + SF + RA +
## CG + SHO + SV + IPouts + HRA + BBA + E + FP + BPF + PPF +
## salary + GIDP + BABIP + RC, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9901 -1.9391 -0.1523 1.9737 7.8582
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1224.786161854707 345.895721393242 3.541 0.000446 ***
## R 0.080154881164 0.006191576342 12.946 < 2e-16 ***
## AB -0.115370180878 0.014302511665 -8.066 8.71e-15 ***
## H 0.294198894471 0.049836065902 5.903 7.66e-09 ***
## X2B 0.034434010453 0.010055404520 3.424 0.000680 ***
## BB 0.029747234634 0.006800119630 4.375 1.56e-05 ***
## SO 0.048815441474 0.009683749780 5.041 7.06e-07 ***
## CS -0.044627799778 0.015532577786 -2.873 0.004283 **
## SF -0.096303670130 0.021621972055 -4.454 1.10e-05 ***
## RA -0.057347471123 0.004031932044 -14.223 < 2e-16 ***
## CG 0.154701717514 0.046819638380 3.304 0.001039 **
## SHO 0.143225421132 0.048746440530 2.938 0.003495 **
## SV 0.346070608015 0.025076294754 13.801 < 2e-16 ***
## IPouts 0.058770788132 0.007084174725 8.296 1.71e-15 ***
## HRA -0.022515047089 0.008581255343 -2.624 0.009033 **
## BBA -0.008098302987 0.002628330947 -3.081 0.002206 **
## E -0.172337651761 0.056648920630 -3.042 0.002505 **
## FP -977.032306332758 348.883980530369 -2.800 0.005353 **
## BPF -0.639264997534 0.233137357795 -2.742 0.006383 **
## PPF 0.599162430181 0.234965627610 2.550 0.011148 *
## salary 0.000000008060 0.000000004559 1.768 0.077838 .
## GIDP -0.047142718419 0.012117297039 -3.891 0.000117 ***
## BABIP -723.253375876375 138.139293383544 -5.236 2.67e-07 ***
## RC -0.109319751926 0.024176237333 -4.522 8.11e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.704 on 396 degrees of freedom
## Multiple R-squared: 0.9489, Adjusted R-squared: 0.9459
## F-statistic: 319.4 on 23 and 396 DF, p-value: < 2.2e-16Step Search: BIC Model
### Step Search: BIC Model
step_bic_model <- step(step_search_start_model, scope = scope, direction = "both",
k = log(n), trace = 0)
summary(step_bic_model)##
## Call:
## lm(formula = W ~ SV + RA + CG + OBP + X3B + SHO + R, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7419 -1.9652 0.0052 1.8342 9.5000
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.123870 6.025094 6.493 0.000000000242 ***
## SV 0.427352 0.025372 16.844 < 2e-16 ***
## RA -0.076829 0.002616 -29.364 < 2e-16 ***
## CG 0.152982 0.046478 3.292 0.001082 **
## OBP 65.425764 23.940331 2.733 0.006549 **
## X3B -0.063520 0.016876 -3.764 0.000192 ***
## SHO 0.174222 0.053140 3.279 0.001132 **
## R 0.079990 0.004115 19.439 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.01 on 412 degrees of freedom
## Multiple R-squared: 0.9341, Adjusted R-squared: 0.9329
## F-statistic: 833.7 on 7 and 412 DF, p-value: < 2.2e-16Step Search: AIC Model
### Step Search: AIC Model ###
step_aic_model <- step(step_search_start_model, scope = scope,
direction = "both", trace = 0)
summary(step_aic_model)##
## Call:
## lm(formula = W ~ SV + RA + CG + X3B + SHO + R + IPouts + AB +
## H + CS + GIDP + SF + BBA + HRA + E + FP + BPF + PPF + salary +
## HA, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.5195 -1.9115 -0.1021 1.8888 8.0195
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 914.034003144572 347.045241981607 2.634 0.008773 **
## SV 0.355614506804 0.024640390657 14.432 < 2e-16 ***
## RA -0.050247955833 0.006340159551 -7.925 0.000000000000023 ***
## CG 0.153017526099 0.044231471311 3.459 0.000600 ***
## X3B -0.068574679286 0.016831959357 -4.074 0.000055748930342 ***
## SHO 0.138199572874 0.049132489791 2.813 0.005154 **
## R 0.084827624066 0.003584776861 23.663 < 2e-16 ***
## IPouts 0.061359854959 0.006880713970 8.918 < 2e-16 ***
## AB -0.052857166713 0.005932828505 -8.909 < 2e-16 ***
## H 0.049033080922 0.006741047553 7.274 0.000000000001865 ***
## CS -0.046638144061 0.015078914455 -3.093 0.002121 **
## GIDP -0.043755357335 0.011669100346 -3.750 0.000203 ***
## SF -0.052733758025 0.019580831875 -2.693 0.007377 **
## BBA -0.010442337072 0.003185502948 -3.278 0.001137 **
## HRA -0.021622522454 0.008707631554 -2.483 0.013432 *
## E -0.161327447529 0.057050198834 -2.828 0.004923 **
## FP -884.643492393877 352.491658585801 -2.510 0.012479 *
## BPF -0.629501587254 0.230983379839 -2.725 0.006707 **
## PPF 0.584792736874 0.231600448463 2.525 0.011956 *
## salary 0.000000007730 0.000000004537 1.704 0.089183 .
## HA -0.005934333997 0.004089404824 -1.451 0.147524
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.727 on 399 degrees of freedom
## Multiple R-squared: 0.9476, Adjusted R-squared: 0.945
## F-statistic: 360.7 on 20 and 399 DF, p-value: < 2.2e-16Forward Search: BIC Model
bic_forward_model <- step(step_search_start_model, scope = scope, k = log(n),trace = 0)
summary(bic_forward_model)##
## Call:
## lm(formula = W ~ SV + RA + CG + OBP + X3B + SHO + R, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7419 -1.9652 0.0052 1.8342 9.5000
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.123870 6.025094 6.493 0.000000000242 ***
## SV 0.427352 0.025372 16.844 < 2e-16 ***
## RA -0.076829 0.002616 -29.364 < 2e-16 ***
## CG 0.152982 0.046478 3.292 0.001082 **
## OBP 65.425764 23.940331 2.733 0.006549 **
## X3B -0.063520 0.016876 -3.764 0.000192 ***
## SHO 0.174222 0.053140 3.279 0.001132 **
## R 0.079990 0.004115 19.439 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.01 on 412 degrees of freedom
## Multiple R-squared: 0.9341, Adjusted R-squared: 0.9329
## F-statistic: 833.7 on 7 and 412 DF, p-value: < 2.2e-16Forward Search: AIC Model
aic_forward_model <- step(step_search_start_model, scope = scope, trace = 0)
summary(aic_forward_model)##
## Call:
## lm(formula = W ~ SV + RA + CG + X3B + SHO + R + IPouts + AB +
## H + CS + GIDP + SF + BBA + HRA + E + FP + BPF + PPF + salary +
## HA, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.5195 -1.9115 -0.1021 1.8888 8.0195
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 914.034003144572 347.045241981607 2.634 0.008773 **
## SV 0.355614506804 0.024640390657 14.432 < 2e-16 ***
## RA -0.050247955833 0.006340159551 -7.925 0.000000000000023 ***
## CG 0.153017526099 0.044231471311 3.459 0.000600 ***
## X3B -0.068574679286 0.016831959357 -4.074 0.000055748930342 ***
## SHO 0.138199572874 0.049132489791 2.813 0.005154 **
## R 0.084827624066 0.003584776861 23.663 < 2e-16 ***
## IPouts 0.061359854959 0.006880713970 8.918 < 2e-16 ***
## AB -0.052857166713 0.005932828505 -8.909 < 2e-16 ***
## H 0.049033080922 0.006741047553 7.274 0.000000000001865 ***
## CS -0.046638144061 0.015078914455 -3.093 0.002121 **
## GIDP -0.043755357335 0.011669100346 -3.750 0.000203 ***
## SF -0.052733758025 0.019580831875 -2.693 0.007377 **
## BBA -0.010442337072 0.003185502948 -3.278 0.001137 **
## HRA -0.021622522454 0.008707631554 -2.483 0.013432 *
## E -0.161327447529 0.057050198834 -2.828 0.004923 **
## FP -884.643492393877 352.491658585801 -2.510 0.012479 *
## BPF -0.629501587254 0.230983379839 -2.725 0.006707 **
## PPF 0.584792736874 0.231600448463 2.525 0.011956 *
## salary 0.000000007730 0.000000004537 1.704 0.089183 .
## HA -0.005934333997 0.004089404824 -1.451 0.147524
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.727 on 399 degrees of freedom
## Multiple R-squared: 0.9476, Adjusted R-squared: 0.945
## F-statistic: 360.7 on 20 and 399 DF, p-value: < 2.2e-16Evaluation and Refinement of the Candidate Models
Evaluation of the Backwards Search: BIC Model
### Breusch-Pagan Test on Backwards Search - BIC Model
library(lmtest)
bptest(bic_model)##
## studentized Breusch-Pagan test
##
## data: bic_model
## BP = 19.912, df = 20, p-value = 0.4635plot_fitted_versus_residuals(fitted(bic_model),
resid(bic_model),
"Backwards Search - BIC Model")
### Shapiro - Wilk Normality Test on Backwards Search - BIC Model ###
shapiro.test(resid(bic_model))##
## Shapiro-Wilk normality test
##
## data: resid(bic_model)
## W = 0.99328, p-value = 0.05825qqnorm(resid(bic_model),
main = "Normal Q-Q Plot, Backwards Search - BIC Model",
col = "darkgrey")
qqline(resid(bic_model), col = "dodgerblue", lwd = 2)
print(paste("Leave One Out Cross-Validated RMSE: ",
round(calc_loocv_rmse(bic_model), 2)))## [1] "Leave One Out Cross-Validated RMSE: 2.81"### Do the number of standard residuals greater than 2 exceed 5% of the total
### observations -- Backwards Search: BIC Model
std_resid_bic_model <- rstandard(bic_model)[abs(rstandard(bic_model)) > 2]
is_std_resid_gt_five_percent_bic_model <- length(std_resid_bic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent_bic_model,
"Outliers Exceed 5% of Obs",
"Outliers Do Not Exceed 5% of Obs")## [1] "Outliers Do Not Exceed 5% of Obs"### VIF > 5 for Backwards Search: BIC Model Coefficients
vif_bic_model <- vif(bic_model)
vif_bic_model[which(vif_bic_model > 5)]## R AB H BB SO RA E FP BABIP RC
## 14.894205 72.403111 929.925966 12.247576 80.989904 6.226679 48.143435 48.871767 122.560946 226.450179Refining the Backwards Search: BIC Model Due to High Variance Inflation Factors
library(caret)bic_model_high_vif_cols <- c("R", "AB", "H", "BB", "SO", "RA", "E", "FP",
"BABIP", "RC")
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(bic_model_high_vif_cols)]),
cutoff = 0.6)
(vars_to_drop <- bic_model_high_vif_cols[indices_to_drop])## [1] "H" "RC" "R" "FP"A Better And Smaller Backwards Search: BIC Model
smaller_bic_model <- lm(W ~ AB + BB + SO + SF + RA + CG + SV + IPouts + BBA + BABIP, data = bbproj_trn)
summary(smaller_bic_model)##
## Call:
## lm(formula = W ~ AB + BB + SO + SF + RA + CG + SV + IPouts +
## BBA + BABIP, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.0090 -3.1465 0.0746 3.1017 13.2658
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -61.891791 33.514874 -1.847 0.065513 .
## AB 0.035857 0.005639 6.358 0.000000000546 ***
## BB 0.055611 0.003754 14.812 < 2e-16 ***
## SO -0.008264 0.002508 -3.296 0.001067 **
## SF 0.083700 0.033456 2.502 0.012746 *
## RA -0.070294 0.004609 -15.250 < 2e-16 ***
## CG 0.335688 0.080389 4.176 0.000036315586 ***
## SV 0.599497 0.041742 14.362 < 2e-16 ***
## IPouts -0.019466 0.010220 -1.905 0.057516 .
## BBA -0.008600 0.004530 -1.898 0.058341 .
## BABIP 118.264318 33.616837 3.518 0.000484 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.888 on 409 degrees of freedom
## Multiple R-squared: 0.8274, Adjusted R-squared: 0.8231
## F-statistic: 196 on 10 and 409 DF, p-value: < 2.2e-16Diagnostics for the Better and Smaller Backwards Search: BIC Model
model_diagnostics(smaller_bic_model)## [1] "Variance Inflation Factors"
## AB BB SO SF RA CG SV IPouts BBA BABIP
## 3.480184 1.210000 1.693955 1.472823 2.839735 1.243693 1.573034 2.843318 1.558630 2.279289
## [1] "Number of coefficients with VIF > 5: 0"
##
## studentized Breusch-Pagan test
##
## data: model
## BP = 13.488, df = 10, p-value = 0.1976
##
##
## Shapiro-Wilk normality test
##
## data: resid(model)
## W = 0.99706, p-value = 0.6567
##
## [1] "Leave One Out Cross-Validated RMSE: 4.95"
## [1] "Outliers Exceed 5% of Obs"Evaluation of the Backwards Search: AIC Model
### Breusch-Pagan Test on AIC Model
bptest(aic_model)##
## studentized Breusch-Pagan test
##
## data: aic_model
## BP = 22.713, df = 23, p-value = 0.4777plot_fitted_versus_residuals(fitted(aic_model), resid(aic_model),
"Backwards Search - AIC Model")
### Shapiro - Wilk Normality Test on Backwards Search - AIC Model ###
shapiro.test(resid(aic_model))##
## Shapiro-Wilk normality test
##
## data: resid(aic_model)
## W = 0.99355, p-value = 0.07005qqnorm(resid(aic_model),
main = "Normal Q-Q Plot, Backwards Search - AIC Model",
col = "darkgrey")
qqline(resid(aic_model), col = "dodgerblue", lwd = 2)
## [1] "Leave One Out Cross-Validated RMSE: 2.79"### Do the number of standard residuals greater than 2 exceed 5% of the total
### observations -- Backwards Search: AIC Model
std_resid_aic_model <- rstandard(aic_model)[abs(rstandard(aic_model)) > 2]
is_std_resid_gt_five_percent_aic_model <- length(std_resid_aic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent_aic_model,
"Outliers Exceed 5% of Obs",
"Outliers Do Not Exceed 5% of Obs")## [1] "Outliers Do Not Exceed 5% of Obs"### VIF > 5 for Backwards Search: AIC Model Coefficients
vif_aic_model <- vif(aic_model)
vif_aic_model[which(vif_aic_model > 5)]## R AB H BB SO RA E FP BPF PPF BABIP RC
## 15.427663 73.154978 950.861657 12.972390 82.555959 7.100542 50.740746 51.448558 85.281044 84.114946 125.780508 232.943354Refining the Backwards Search: AIC Model Due to High Variance Inflation Factors
aic_model_high_vif_cols <- c("R", "AB", "H", "X2B", "X3B", "HR", "BB", "SO",
"SF", "RA", "ER", "IPouts", "E", "FP","BPF", "PPF",
"IBB", "OBP", "BABIP", "RC", "wOBA")
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(aic_model_high_vif_cols)]),
cutoff = 0.6)
(vars_to_drop <- aic_model_high_vif_cols[indices_to_drop])## [1] "RC" "wOBA" "R" "OBP" "H" "RA" "BPF" "FP"A Better And Smaller Backwards Search: AIC Model
# smaller_aic_model <- lm(formula = W ~ AB + HR + BB + SO + CS + ER + CG +
# SHO + SV + IPouts + BBA + FP + GIDP + IBB + BABIP,
# data = bbproj)
smaller_aic_model <- lm(formula = W ~ HR + BB + SO + ER + CG + SHO + SV +
IPouts + BBA + FP + GIDP + BABIP, data = bbproj_trn)
summary(smaller_aic_model)##
## Call:
## lm(formula = W ~ HR + BB + SO + ER + CG + SHO + SV + IPouts +
## BBA + FP + GIDP + BABIP, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.3371 -2.5236 -0.0841 2.1182 10.2418
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -328.428863 67.755211 -4.847 0.00000178 ***
## HR 0.132223 0.005998 22.044 < 2e-16 ***
## BB 0.032238 0.002798 11.523 < 2e-16 ***
## SO -0.023360 0.001674 -13.952 < 2e-16 ***
## ER -0.068993 0.003633 -18.993 < 2e-16 ***
## CG 0.189266 0.057990 3.264 0.001192 **
## SHO 0.235242 0.061006 3.856 0.000134 ***
## SV 0.410129 0.030171 13.593 < 2e-16 ***
## IPouts 0.014815 0.005185 2.857 0.004495 **
## BBA -0.008063 0.003203 -2.517 0.012216 *
## FP 278.765319 69.652956 4.002 0.00007457 ***
## GIDP -0.037475 0.012793 -2.929 0.003589 **
## BABIP 316.088714 16.896493 18.707 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.435 on 407 degrees of freedom
## Multiple R-squared: 0.9152, Adjusted R-squared: 0.9127
## F-statistic: 365.9 on 12 and 407 DF, p-value: < 2.2e-16Diagnostics for the Better and Smaller Backwards Search: AIC Model
model_diagnostics(smaller_aic_model)## [1] "Variance Inflation Factors"
## HR BB SO ER CG SHO SV IPouts BBA FP GIDP BABIP
## 1.426241 1.360620 1.529067 3.064298 1.310541 1.907533 1.664221 1.482259 1.578259 1.270639 1.409269 1.166015
## [1] "Number of coefficients with VIF > 5: 0"
##
## studentized Breusch-Pagan test
##
## data: model
## BP = 8.7708, df = 12, p-value = 0.7224
##
##
## Shapiro-Wilk normality test
##
## data: resid(model)
## W = 0.99241, p-value = 0.03137
##
## [1] "Leave One Out Cross-Validated RMSE: 3.49"
## [1] "Outliers Exceed 5% of Obs"Evaluation of the Step Search: BIC Model
### Breusch-Pagan Test on Step BIC Model
bptest(step_bic_model)##
## studentized Breusch-Pagan test
##
## data: step_bic_model
## BP = 3.0387, df = 7, p-value = 0.8814plot_fitted_versus_residuals(fitted(step_bic_model), resid(step_bic_model),
"Step Search - BIC Model")
### Shapiro - Wilk Normality Test on Step Search - BIC Model ###
shapiro.test(resid(step_bic_model))##
## Shapiro-Wilk normality test
##
## data: resid(step_bic_model)
## W = 0.99708, p-value = 0.6604qqnorm(resid(step_bic_model),
main = "Normal Q-Q Plot, Step Search - BIC Model",
col = "darkgrey")
qqline(resid(step_bic_model), col = "dodgerblue", lwd = 2)
## [1] "Leave One Out Cross-Validated RMSE: 3.04"### Do the number of standard residuals greater than 2 exceed 5% of the total
### observations -- Step Search: BIC Model
std_resid_step_bic_model <- rstandard(
step_bic_model)[abs(rstandard(step_bic_model)) > 2]
is_std_resid_gt_five_percent__step_bic_model <- length(
std_resid_step_bic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent__step_bic_model, "Exceeds 5% of Obs",
"Does Not Exceed 5% of Obs")## [1] "Does Not Exceed 5% of Obs"### VIF > 5 for Step Search: BIC Model Coefficients
vif_step_bic_model <- vif(step_bic_model)
vif_step_bic_model[which(vif_step_bic_model > 5)]## OBP R
## 5.221787 5.499516Refining the Step Search: BIC Model Due to High Variance Inflation Factors
A Better And Smaller Step Search: BIC Model
smaller_step_bic_model <- lm(formula = W ~ SV + R + RA + SHO + CG + X3B +
IPouts + AB + salary, data = bbproj_trn)
summary(smaller_step_bic_model)##
## Call:
## lm(formula = W ~ SV + R + RA + SHO + CG + X3B + IPouts + AB +
## salary, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.6930 -2.0156 0.0027 1.8031 8.6432
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.64791877157 19.43171439428 0.857 0.39209
## SV 0.39794921940 0.02548723971 15.614 < 2e-16 ***
## R 0.09465788997 0.00239639476 39.500 < 2e-16 ***
## RA -0.07000030397 0.00295754734 -23.668 < 2e-16 ***
## SHO 0.17217667234 0.05222372960 3.297 0.00106 **
## CG 0.14266098090 0.04629864831 3.081 0.00220 **
## X3B -0.04901240782 0.01691431490 -2.898 0.00396 **
## IPouts 0.02307347246 0.00528945937 4.362 0.0000163 ***
## AB -0.01298730710 0.00293489287 -4.425 0.0000124 ***
## salary 0.00000001035 0.00000000449 2.305 0.02167 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.95 on 410 degrees of freedom
## Multiple R-squared: 0.937, Adjusted R-squared: 0.9356
## F-statistic: 677.2 on 9 and 410 DF, p-value: < 2.2e-16Diagnostics for the Better and Smaller Step Search: BIC Model
model_diagnostics(smaller_step_bic_model)## [1] "Variance Inflation Factors"
## SV R RA SHO CG X3B IPouts AB salary
## 1.610244 1.941649 3.209850 1.895339 1.132668 1.067672 2.091258 2.587975 1.267406
## [1] "Number of coefficients with VIF > 5: 0"
##
## studentized Breusch-Pagan test
##
## data: model
## BP = 4.372, df = 9, p-value = 0.8853
##
##
## Shapiro-Wilk normality test
##
## data: resid(model)
## W = 0.9967, p-value = 0.549
##
## [1] "Leave One Out Cross-Validated RMSE: 2.99"
## [1] "Outliers Do Not Exceed 5% of Obs"Evaluation of the Step Search: AIC Model
### Breusch-Pagan Test on Step AIC Model
bptest(step_aic_model)##
## studentized Breusch-Pagan test
##
## data: step_aic_model
## BP = 22.885, df = 20, p-value = 0.2945plot_fitted_versus_residuals(fitted(step_aic_model), resid(step_aic_model),
"Step Search - AIC Model")
### Shapiro - Wilk Normality Test on Step Search - AIC Model ###
shapiro.test(resid(step_aic_model))##
## Shapiro-Wilk normality test
##
## data: resid(step_aic_model)
## W = 0.99683, p-value = 0.5883qqnorm(resid(step_aic_model),
main = "Normal Q-Q Plot, Step Search - AIC Model",
col = "darkgrey")
qqline(resid(step_aic_model), col = "dodgerblue", lwd = 2)
## [1] "Leave One Out Cross-Validated RMSE: 2.8"### Do the number of standard residuals greater than 2 exceed 5% of the total
### observations -- Step Search: AIC Model
std_resid_step_aic_model <- rstandard(
step_aic_model)[abs(rstandard(step_aic_model)) > 2]
is_std_resid_gt_five_percent__step_aic_model <- length(
std_resid_step_aic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent__step_aic_model,
"Exceeds 5% of Obs",
"Does Not Exceed 5% of Obs")## [1] "Does Not Exceed 5% of Obs"### VIF > 5 for Step Search: AIC Model Coefficients
vif_step_aic_model <- vif(step_aic_model)
vif_step_aic_model[which(vif_step_aic_model > 5)]## RA R AB H E FP BPF PPF HA
## 17.262491 5.084637 12.376016 17.104981 50.597130 51.635315 82.305380 80.349149 7.123143Refining the Step Search: AIC Model Due to High Variance Inflation Factors
A Better And Smaller Step Search: AIC Model
smaller_step_aic_model <- lm(formula = W ~ SV + R + SHO + CG + X3B + IPouts +
BBA + HRA + SOA, data = bbproj_trn)
summary(smaller_step_aic_model)##
## Call:
## lm(formula = W ~ SV + R + SHO + CG + X3B + IPouts + BBA + HRA +
## SOA, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.9548 -2.9275 0.2111 2.5960 15.3693
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -106.910065 24.782771 -4.314 0.000020123812728 ***
## SV 0.553970 0.033062 16.756 < 2e-16 ***
## R 0.082172 0.002520 32.612 < 2e-16 ***
## SHO 0.459052 0.068216 6.729 0.000000000057562 ***
## CG 0.295343 0.063584 4.645 0.000004589228066 ***
## X3B -0.063341 0.022615 -2.801 0.00534 **
## IPouts 0.026697 0.005825 4.583 0.000006087870449 ***
## BBA -0.027269 0.003450 -7.905 0.000000000000025 ***
## HRA -0.098460 0.009868 -9.978 < 2e-16 ***
## SOA 0.013936 0.001999 6.970 0.000000000012736 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.006 on 410 degrees of freedom
## Multiple R-squared: 0.8837, Adjusted R-squared: 0.8812
## F-statistic: 346.3 on 9 and 410 DF, p-value: < 2.2e-16Diagnostics for the Better and Smaller Step Search: AIC Model
model_diagnostics(smaller_step_aic_model)## [1] "Variance Inflation Factors"
## SV R SHO CG X3B IPouts BBA HRA SOA
## 1.468999 1.163807 1.753261 1.158178 1.034795 1.375007 1.345455 1.577315 1.440348
## [1] "Number of coefficients with VIF > 5: 0"
##
## studentized Breusch-Pagan test
##
## data: model
## BP = 3.5705, df = 9, p-value = 0.9373
##
##
## Shapiro-Wilk normality test
##
## data: resid(model)
## W = 0.99481, p-value = 0.1694
##
## [1] "Leave One Out Cross-Validated RMSE: 4.05"
## [1] "Outliers Do Not Exceed 5% of Obs"Evaluation of the Forward Search: BIC Model
### Breusch-Pagan Test
bptest(bic_forward_model)##
## studentized Breusch-Pagan test
##
## data: bic_forward_model
## BP = 3.0387, df = 7, p-value = 0.8814plot_fitted_versus_residuals(fitted(bic_forward_model), resid(bic_forward_model), "Forward Search - BIC Model")
### Shapiro - Wilk Normality Test ###
shapiro.test(resid(bic_forward_model))##
## Shapiro-Wilk normality test
##
## data: resid(bic_forward_model)
## W = 0.99708, p-value = 0.6604qqnorm(resid(bic_forward_model), main = "Normal Q-Q Plot, Step Search - BIC Model", col = "darkgrey")
qqline(resid(bic_forward_model), col = "dodgerblue", lwd = 2)
## [1] "Leave One Out Cross-Validated RMSE: 3.04"### Do the number of standard residuals greater than 2 exceed 5% of the total observations -- Step Search: AIC Model
std_resid <- rstandard(bic_forward_model)[abs(rstandard(bic_forward_model)) > 2]
is_std_resid_gt_five_percent <- length(std_resid) / n > 0.05
ifelse(is_std_resid_gt_five_percent,"Exceeds 5% of Obs", "Does Not Exceed 5% of Obs")## [1] "Does Not Exceed 5% of Obs"### VIF > 5 for Step Search: AIC Model Coefficients
library(faraway)
vifs <- vif(bic_forward_model)
vifs[which(vifs > 5)]## OBP R
## 5.221787 5.499516Refining the Forward Search: BIC Model Due to High Variance Inflation Factors
library(caret)
high_vif_cols <- as.array(names(vifs[which(vifs > 5)]))
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(high_vif_cols)]), cutoff = 0.6)
vars_to_drop <- high_vif_cols[indices_to_drop]
vars_to_drop## [1] "R"A Better And Smaller Forward Search: BIC Model
terms = attr(bic_forward_model$terms, "term.labels")
terms = terms[! terms %in% vars_to_drop]
formula = as.formula(paste("W ~", paste(terms, collapse = "+")))
smaller_model <- lm(formula = formula, data = bbproj_trn)
summary(smaller_model)##
## Call:
## lm(formula = formula, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.2187 -2.7593 -0.0139 2.6440 11.9379
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -48.379453 5.538267 -8.735 < 2e-16 ***
## SV 0.510814 0.034581 14.771 < 2e-16 ***
## RA -0.067633 0.003559 -19.005 < 2e-16 ***
## CG 0.259678 0.063826 4.069 0.0000567 ***
## OBP 478.519624 15.245069 31.388 < 2e-16 ***
## X3B -0.058658 0.023336 -2.514 0.0123 *
## SHO 0.166892 0.073487 2.271 0.0237 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.162 on 413 degrees of freedom
## Multiple R-squared: 0.8736, Adjusted R-squared: 0.8717
## F-statistic: 475.7 on 6 and 413 DF, p-value: < 2.2e-16smaller_bic_forward_model = smaller_modelDiagnostics for the Better and Smaller Forward Search: BIC Model
model_diagnostics(smaller_bic_forward_model)## [1] "Variance Inflation Factors"
## SV RA CG OBP X3B SHO
## 1.488879 2.334176 1.081155 1.107182 1.020757 1.884954
## [1] "Number of coefficients with VIF > 5: 0"
##
## studentized Breusch-Pagan test
##
## data: model
## BP = 3.5489, df = 6, p-value = 0.7375
##
##
## Shapiro-Wilk normality test
##
## data: resid(model)
## W = 0.99807, p-value = 0.9205
##
## [1] "Leave One Out Cross-Validated RMSE: 4.2"
## [1] "Outliers Do Not Exceed 5% of Obs"Evaluation of the Forward Search: AIC Model
### Breusch-Pagan Test
bptest(aic_forward_model)##
## studentized Breusch-Pagan test
##
## data: aic_forward_model
## BP = 22.885, df = 20, p-value = 0.2945plot_fitted_versus_residuals(fitted(aic_forward_model), resid(aic_forward_model), "Forward Search - AIC Model")
### Shapiro - Wilk Normality Test ###
shapiro.test(resid(aic_forward_model))##
## Shapiro-Wilk normality test
##
## data: resid(aic_forward_model)
## W = 0.99683, p-value = 0.5883qqnorm(resid(aic_forward_model), main = "Normal Q-Q Plot, Step Search - AIC Model", col = "darkgrey")
qqline(resid(aic_forward_model), col = "dodgerblue", lwd = 2)
## [1] "Leave One Out Cross-Validated RMSE: 2.8"### Do the number of standard residuals greater than 2 exceed 5% of the total observations -- Step Search: AIC Model
std_resid <- rstandard(aic_forward_model)[abs(rstandard(aic_forward_model)) > 2]
is_std_resid_gt_five_percent <- length(std_resid) / n > 0.05
ifelse(is_std_resid_gt_five_percent,"Exceeds 5% of Obs", "Does Not Exceed 5% of Obs")## [1] "Does Not Exceed 5% of Obs"### VIF > 5 for Step Search: AIC Model Coefficients
library(faraway)
vifs <- vif(aic_forward_model)
vifs[which(vifs > 5)]## RA R AB H E FP BPF PPF HA
## 17.262491 5.084637 12.376016 17.104981 50.597130 51.635315 82.305380 80.349149 7.123143Refining the Forward Search: AIC Model Due to High Variance Inflation Factors
library(caret)
high_vif_cols <- as.array(names(vifs[which(vifs > 5)]))
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(high_vif_cols)]), cutoff = 0.6)
vars_to_drop <- high_vif_cols[indices_to_drop]
vars_to_drop## [1] "H" "RA" "R" "BPF" "FP"A Better And Smaller Forward Search: AIC Model
terms = attr(aic_forward_model$terms, "term.labels")
terms = terms[! terms %in% vars_to_drop]
formula = as.formula(paste("W ~", paste(terms, collapse = "+")))
smaller_model <- lm(formula = formula, data = bbproj_trn)
summary(smaller_model)##
## Call:
## lm(formula = formula, data = bbproj_trn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20.7707 -4.0132 0.0735 3.9381 16.8375
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -130.372691160134 40.942184890385 -3.184 0.001563 **
## SV 0.691822762359 0.051293290999 13.488 < 2e-16 ***
## CG 0.451451517680 0.097869852286 4.613 0.00000534151951 ***
## X3B -0.044130287173 0.037206097668 -1.186 0.236278
## SHO 0.187051414650 0.110061882108 1.700 0.089993 .
## IPouts 0.006637885812 0.010439115229 0.636 0.525223
## AB 0.035276686983 0.005368182209 6.571 0.00000000015396 ***
## CS -0.018371247437 0.029837527171 -0.616 0.538433
## GIDP 0.026472866957 0.020953425350 1.263 0.207169
## SF 0.273938786852 0.038676525292 7.083 0.00000000000631 ***
## BBA -0.020850806318 0.005466335198 -3.814 0.000158 ***
## HRA -0.033796874832 0.016267982447 -2.078 0.038386 *
## E -0.045858530274 0.020678376355 -2.218 0.027131 *
## PPF 0.154768090993 0.063792547637 2.426 0.015698 *
## salary 0.000000022080 0.000000009774 2.259 0.024405 *
## HA -0.037682947274 0.005416312830 -6.957 0.00000000001406 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.152 on 404 degrees of freedom
## Multiple R-squared: 0.7299, Adjusted R-squared: 0.7198
## F-statistic: 72.77 on 15 and 404 DF, p-value: < 2.2e-16smaller_aic_forward_model = smaller_modelDiagnostics for the Better and Smaller Forward Search: AIC Model
model_diagnostics(smaller_aic_forward_model)## [1] "Variance Inflation Factors"
## SV CG X3B SHO IPouts AB CS GIDP SF BBA HRA E PPF salary
## 1.499417 1.163642 1.187716 1.935442 1.872698 1.990606 1.231270 1.178478 1.242515 1.432725 1.817827 1.305924 1.197612 1.380530
## HA
## 2.454893
## [1] "Number of coefficients with VIF > 5: 0"
##
## studentized Breusch-Pagan test
##
## data: model
## BP = 20.617, df = 15, p-value = 0.1495
##
##
## Shapiro-Wilk normality test
##
## data: resid(model)
## W = 0.9972, p-value = 0.6962
##
## [1] "Leave One Out Cross-Validated RMSE: 6.27"
## [1] "Outliers Do Not Exceed 5% of Obs"Validate Model Effectiveness Using Test Data
Results
And the Winning Linear Regression Model is…
Discussion
Logistic Modeling and Prediction of Division Winners
Salary as a Predictor for Division Winners
Let’s turn our attention to logistic regression and our ability to classify and predict division winners using our team predictor set. First, let’s see how well salary alone can predict pennants.
salary_only_model <- glm(DivWin ~ salary, data = bbproj_trn, family = binomial)
(salary_only_summary = summary(salary_only_model))##
## Call:
## glm(formula = DivWin ~ salary, family = binomial, data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5538 -0.6746 -0.5664 -0.4673 2.1545
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.693550028350 0.317777033524 -8.476 < 2e-16 ***
## salary 0.000000015283 0.000000003249 4.704 0.00000255 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 397.38 on 418 degrees of freedom
## AIC: 401.38
##
## Number of Fisher Scoring iterations: 4plot(as.numeric(DivWin) - 1 ~ salary, data = bbproj_trn,
pch = 20,
main = "Probability of Buying a Division Winner",
ylab = "Probability of Winning Division",
xlab = "Team Salary ($)",
xlim = c(0, 3e8))
curve(predict(salary_only_model, data.frame(salary = x), type = "response"),
add = TRUE,
col = "tomato",
lty = 2,
lwd = 2)
Graphically, a team’s payroll seems to be moderately influential in predicting their chances of taking home a division crown. Indeed, our Wald test for salary alone yields a p-value of 0.0000025, allowing us to reject the null hypothesis (\(H_0 : \beta_{salary} = 0\)) for any reasonable value for \(\alpha\). So what happens when we look at our salary model’s misclassification rate?
salary_prediction <- ifelse(predict(salary_only_model,
bbproj_tst,
type = "response") > 0.5, "Y", "N")
(prevalence = table(bbproj_tst$DivWin) / nrow(bbproj_tst))##
## N Y
## 0.8 0.2(salary_misclass = mean(salary_prediction != bbproj_tst$DivWin))## [1] 0.2222222First, let’s examine prevalence of division winners. We see that 20% of the teams were division winners, which makes sense, since there are 6 divisions and 30 MLB teams, therefore you will only have 6 division winners per year. Our salary model has a misclassification rate of 0.222, which is worse than our prevalence. We would have a better misclassification rate if we simply stated that there are no division winners! We can certainly do better than this.
Methodology
Candidate Logistic Regression Models
We will begin our search for a better classifier by setting up an initial logistic regression model contain all of the predictors we would like to evaluate. Then, we will proceed to eliminate predictors using backwards, forwards, and stepwise AIC and BIC searches. See Appendix C for a complete evaluation of all of the models. The following will detail the methodology using our best candidate model.
scope <- DivWin ~ W + R + AB + H + X2B + X3B + HR + BB + SO + SB + CS + HBP +
SF + RA + ER + ERA + CG + SHO + SV + IPouts + HA + HRA + BBA + SOA + E +
DP + FP + attendance + BPF + PPF + salary + RBI + GIDP + IBB + TB + SLG +
OBP + OPS + WHIP + BABIP + RC + X1B + uBB + wOBA
start_model <- glm(DivWin ~ 1, data = bbproj_trn, family = binomial)
n <- length(resid(start_model))Forward Search: BIC Model
bic_model <- step(start_model, direction = "forward", scope = scope,
k = log(n), trace = 0)
summary(bic_model)##
## Call:
## glm(formula = DivWin ~ W + X2B, family = binomial, data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.35987 -0.30139 -0.06081 -0.00867 2.59348
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -27.523331 3.832156 -7.182 0.000000000000686 ***
## W 0.350030 0.042623 8.212 < 2e-16 ***
## X2B -0.016344 0.006758 -2.418 0.0156 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 194.55 on 417 degrees of freedom
## AIC: 200.55
##
## Number of Fisher Scoring iterations: 7Our forward search using BIC yields the following logistic model
\[ \log \bigg(\frac{P[DivWin = 1]}{1 - P[DivWin = 1]} \bigg) = \beta_0 + \beta_W W + \beta_{X2B} X2B \]
where the log odds of a team winning their division is dependent upon wins and doubles.
Diagnostics: BIC Model
Next, we will create a function that will evaluate this model using a confusion matrix and calculate its sensitivity, specificity, and its misclassification rate (the classify_and_diagnose function code can be found in Appendix C). Then we will start by examining a default cutoff value of 0.5.
(bic_diag <- classify_and_diagnose(bic_model))## $confusion_matrix
## actual
## prediction N Y
## N 70 7
## Y 2 11
##
## $sensitivity
## [1] 0.6111111
##
## $specificity
## [1] 0.9722222
##
## $misclassification
## [1] 0.1Here, we beat the salary only model along with the prevalence with a misclassification rate of 0.1. Our specificity looks great for our forward BIC model at 0.972. Our 7 false negatives could come down, however, with an adjustment to our cutoff value.
Find an Optimal Cutoff
Next we will define a function that will loop through a vector of potential cutoffs to isolate one that will produce the smallest misclassification rate with a minimal differential between sensitivity and specificity (see Appendix C for the opt_logistic_cutoff function’s definition).
(opt_cutoff = opt_logistic_cutoff(bic_model, cut_start = 0.1, cut_end = 0.8))
## cutoff misclass sensitivity specificity
## 0.30000000 0.05555556 0.88888889 0.95833333classify_and_diagnose(bic_model, cutoff = opt_cutoff["cutoff"])$confusion_matrix## actual
## prediction N Y
## N 69 2
## Y 3 16Our routine finds an optimal cutoff value for our forward BIC model of 0.3. Our misclassification rate went down with our cutoff adjustment to 0.056. Additionally, our sensitivity went up significantly (0.889) and we only had a modest reduction in specificity (0.958). This model looks to be exceptionally adept at classifying division winners, and salary is not even a predictor!
Results
And the Winning Logistic Regression Model is…
Discussion
Apendices
Appendix A - Logistic Regression Models
Classify and Diagnose Function
classify_and_diagnose = function(model, data = bbproj_tst,
actual = bbproj_tst$DivWin,
pos = "Y", neg = "N", cutoff = 0.5) {
# Generate a classifier given the model, data, cutoff, and positive and
# negative responses
pred = ifelse(predict(model,
data,
type = "response") > cutoff, pos, neg)
# Generate the confusion matrix
conf_mat = table(prediction = pred, actual = actual)
# Calculate sensitivity, specificity, and the misclassification rate and
# return them plus the confusion matrix
list(confusion_matrix = conf_mat,
sensitivity = conf_mat[2, 2] / sum(conf_mat[, 2]),
specificity = conf_mat[1, 1] / sum(conf_mat[, 1]),
misclassification = mean(pred != actual))
}Optimal Logistic Cutoff Function
opt_logistic_cutoff = function(model, cut_start = 0.01, cut_end = 0.99,
data = bbproj_tst, actual = bbproj_tst$DivWin,
pos = "Y", neg = "N", plotit = TRUE) {
# Loop through potential cutoffs from cut_start to cut_end to determine a
# cutoff that produces the lowest misclassification rate with the smallest
# delta between sensitivity and specificity
cutoffs = seq(cut_start, cut_end, by = 0.01)
sens = rep(0, length(cutoffs))
spec = rep(0, length(cutoffs))
misclass = rep(0, length(cutoffs))
delta = rep(0, length(cutoffs))
for (i in 1:length(cutoffs)) {
diagnostics = classify_and_diagnose(model,
data = data,
actual = actual,
pos = pos,
neg = neg,
cutoff = cutoffs[i])
sens[i] = diagnostics$sensitivity
spec[i] = diagnostics$specificity
misclass[i] = diagnostics$misclassification
delta[i] = abs(diagnostics$sensitivity - diagnostics$specificity)
}
# Get the indicies of the smallest misclassification rates
min_misclass = which(misclass == min(misclass))
# Get the smallest delta between sensitivity and specificity at the
# identified misclassification indicies
min_delta = min_misclass[which.min(delta[min_misclass])]
# Plot sensitivity, specificity, and misclassification if requested
if (plotit) {
plot(sens ~ cutoffs,
xlab = "Cutoff",
ylab = "Sensitivity/Specificity",
main = "Sensitivity and Specificity at Varied Cutoffs",
col = "tomato",
type = "b",
ylim = c(0, 1),
pch = 20)
lines(spec ~ cutoffs,
col = "darkslategray4",
type = "b",
pch = 20)
lines(misclass ~ cutoffs,
col = "darkorange",
type = "b",
pch = 20)
abline(v = cutoffs[min_delta], lty = 2)
legend("topright", c("Sensitivity",
"Specificity",
"Misclassification"),
col = c("tomato", "darkslategray4", "darkorange"),
lwd = 1,
pch = 20)
}
# Return the misclassification, sensitivity, and specificity at the optimal
# cutoff value
c(cutoff = cutoffs[min_delta],
misclass = misclass[min_delta],
sensitivity = sens[min_delta],
specificity = spec[min_delta])
}Backward Search: BIC Model
formula <- formula(scope)
back_bic_model <- step(glm(formula, data = bbproj_trn, family = binomial),
direction = "backward", k = log(n), trace = 0)
back_bic_diag <- classify_and_diagnose(back_bic_model)summary(back_bic_model)##
## Call:
## glm(formula = DivWin ~ W + AB + H + HR + BB + HBP + IBB + SLG +
## OBP + wOBA, family = binomial, data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.09816 -0.27642 -0.04275 -0.00332 2.76522
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1052.50217 319.70312 -3.292 0.000994 ***
## W 0.37031 0.04793 7.726 0.0000000000000111 ***
## AB 0.17509 0.05585 3.135 0.001717 **
## H -0.52892 0.16383 -3.228 0.001245 **
## HR 0.09433 0.03603 2.618 0.008842 **
## BB -0.34709 0.10338 -3.357 0.000787 ***
## HBP -0.33911 0.09686 -3.501 0.000463 ***
## IBB -0.30746 0.12140 -2.533 0.011320 *
## SLG 1794.92000 702.65755 2.554 0.010635 *
## OBP 6178.22128 2030.78091 3.042 0.002348 **
## wOBA -5414.02198 2077.07626 -2.607 0.009146 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 179.06 on 409 degrees of freedom
## AIC: 201.06
##
## Number of Fisher Scoring iterations: 8back_bic_diag## $confusion_matrix
## actual
## prediction N Y
## N 70 7
## Y 2 11
##
## $sensitivity
## [1] 0.6111111
##
## $specificity
## [1] 0.9722222
##
## $misclassification
## [1] 0.1Optimal Cutoff
opt_logistic_cutoff(back_bic_model, cut_start = 0.1, cut_end = 0.8)
## cutoff misclass sensitivity specificity
## 0.26000000 0.05555556 0.83333333 0.97222222opt_back_bic_diag$confusion_matrix## actual
## prediction N Y
## N 70 3
## Y 2 15Forward Search: BIC Model
forw_bic_model <- step(start_model, direction = "forward", scope = scope,
k = log(n), trace = 0)
forw_bic_diag <- classify_and_diagnose(forw_bic_model)summary(forw_bic_model)##
## Call:
## glm(formula = DivWin ~ W + X2B, family = binomial, data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.35987 -0.30139 -0.06081 -0.00867 2.59348
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -27.523331 3.832156 -7.182 0.000000000000686 ***
## W 0.350030 0.042623 8.212 < 2e-16 ***
## X2B -0.016344 0.006758 -2.418 0.0156 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 194.55 on 417 degrees of freedom
## AIC: 200.55
##
## Number of Fisher Scoring iterations: 7forw_bic_diag## $confusion_matrix
## actual
## prediction N Y
## N 70 7
## Y 2 11
##
## $sensitivity
## [1] 0.6111111
##
## $specificity
## [1] 0.9722222
##
## $misclassification
## [1] 0.1Optimal Cutoff
opt_logistic_cutoff(forw_bic_model, cut_start = 0.1, cut_end = 0.8)
## cutoff misclass sensitivity specificity
## 0.30000000 0.05555556 0.88888889 0.95833333opt_forw_bic_diag$confusion_matrix## actual
## prediction N Y
## N 69 2
## Y 3 16Stepwise Search: BIC Model
step_bic_model <- step(start_model, direction = "both", scope = scope,
k = log(n), trace = 0)
step_bic_diag <- classify_and_diagnose(step_bic_model)summary(step_bic_model)##
## Call:
## glm(formula = DivWin ~ W + X2B, family = binomial, data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.35987 -0.30139 -0.06081 -0.00867 2.59348
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -27.523331 3.832156 -7.182 0.000000000000686 ***
## W 0.350030 0.042623 8.212 < 2e-16 ***
## X2B -0.016344 0.006758 -2.418 0.0156 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 194.55 on 417 degrees of freedom
## AIC: 200.55
##
## Number of Fisher Scoring iterations: 7step_bic_diag## $confusion_matrix
## actual
## prediction N Y
## N 70 7
## Y 2 11
##
## $sensitivity
## [1] 0.6111111
##
## $specificity
## [1] 0.9722222
##
## $misclassification
## [1] 0.1Optimal Cutoff
opt_step_bic <- opt_logistic_cutoff(step_bic_model,
cut_start = 0.1,
cut_end = 0.8,
plotit = FALSE)
opt_step_bic_diag <- classify_and_diagnose(step_bic_model,
cutoff = opt_step_bic["cutoff"])opt_logistic_cutoff(step_bic_model, cut_start = 0.1, cut_end = 0.8)
## cutoff misclass sensitivity specificity
## 0.30000000 0.05555556 0.88888889 0.95833333opt_step_bic_diag$confusion_matrix## actual
## prediction N Y
## N 69 2
## Y 3 16Backward Search: AIC Model
formula <- formula(scope)
back_aic_model <- step(glm(formula, data = bbproj_trn, family = binomial),
direction = "backward", trace = 0)
back_aic_diag <- classify_and_diagnose(back_aic_model)summary(back_aic_model)##
## Call:
## glm(formula = DivWin ~ W + R + AB + H + HR + BB + CS + HBP +
## ER + IPouts + E + FP + BPF + PPF + RBI + IBB + SLG + OBP +
## OPS + wOBA, family = binomial, data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.96822 -0.20843 -0.02111 -0.00065 2.65771
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -140.179255 693.655494 -0.202 0.839848
## W 0.545656 0.090798 6.010 0.00000000186 ***
## R -0.076857 0.035782 -2.148 0.031717 *
## AB 0.183520 0.063041 2.911 0.003601 **
## H -0.592491 0.183941 -3.221 0.001277 **
## HR 0.079762 0.041066 1.942 0.052099 .
## BB -0.401042 0.116424 -3.445 0.000572 ***
## CS -0.059004 0.023336 -2.528 0.011457 *
## HBP -0.393193 0.110648 -3.554 0.000380 ***
## ER 0.024676 0.008386 2.943 0.003255 **
## IPouts 0.022175 0.011255 1.970 0.048806 *
## E -0.168360 0.092043 -1.829 0.067378 .
## FP -1111.221700 583.517366 -1.904 0.056865 .
## BPF 0.736250 0.339446 2.169 0.030084 *
## PPF -0.674039 0.343047 -1.965 0.049431 *
## RBI 0.070973 0.036748 1.931 0.053438 .
## IBB -0.299451 0.134303 -2.230 0.025769 *
## SLG 851959707.701375 417793051.690715 2.039 0.041431 *
## OBP 851964540.159953 417793092.777287 2.039 0.041430 *
## OPS -851957944.218280 417793035.107366 -2.039 0.041431 *
## wOBA -5332.125383 2319.746099 -2.299 0.021529 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 151.22 on 399 degrees of freedom
## AIC: 193.22
##
## Number of Fisher Scoring iterations: 8back_aic_diag <- classify_and_diagnose(back_aic_model)Optimal Cutoff
opt_back_aic <- opt_logistic_cutoff(back_aic_model,
cut_start = 0.1,
cut_end = 0.8,
plotit = FALSE)
opt_back_aic_diag <- classify_and_diagnose(back_aic_model,
cutoff = opt_back_aic["cutoff"])opt_logistic_cutoff(back_aic_model, cut_start = 0.1, cut_end = 0.8)
## cutoff misclass sensitivity specificity
## 0.26000000 0.05555556 0.88888889 0.95833333opt_back_aic_diag$confusion_matrix## actual
## prediction N Y
## N 69 2
## Y 3 16Forward Search: AIC Model
forw_aic_model <- step(start_model, direction = "forward",
scope = scope, trace = 0)
forw_aic_diag <- classify_and_diagnose(forw_aic_model)summary(forw_aic_model)##
## Call:
## glm(formula = DivWin ~ W + X2B + HA + DP + GIDP, family = binomial,
## data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.31485 -0.25596 -0.04893 -0.00565 2.90172
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -37.894889 6.186389 -6.126 9.04e-10 ***
## W 0.392678 0.048963 8.020 1.06e-15 ***
## X2B -0.024866 0.007752 -3.208 0.00134 **
## HA 0.006988 0.002855 2.447 0.01440 *
## DP -0.024295 0.012063 -2.014 0.04401 *
## GIDP 0.021012 0.012242 1.716 0.08609 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 182.92 on 414 degrees of freedom
## AIC: 194.92
##
## Number of Fisher Scoring iterations: 7forw_aic_diag## $confusion_matrix
## actual
## prediction N Y
## N 70 8
## Y 2 10
##
## $sensitivity
## [1] 0.5555556
##
## $specificity
## [1] 0.9722222
##
## $misclassification
## [1] 0.1111111Optimal Cutoff
opt_forw_aic <- opt_logistic_cutoff(forw_aic_model,
cut_start = 0.1,
cut_end = 0.8,
plotit = FALSE)
opt_forw_aic_diag <- classify_and_diagnose(forw_aic_model,
cutoff = opt_forw_aic["cutoff"])opt_logistic_cutoff(forw_aic_model, cut_start = 0.1, cut_end = 0.8)
## cutoff misclass sensitivity specificity
## 0.23000000 0.07777778 0.94444444 0.91666667opt_forw_aic_diag$confusion_matrix## actual
## prediction N Y
## N 66 1
## Y 6 17Stepwise Search: AIC Model
step_aic_model <- step(start_model, direction = "both",
scope = scope, trace = 0)
step_aic_diag <- classify_and_diagnose(step_aic_model)summary(step_aic_model)##
## Call:
## glm(formula = DivWin ~ W + X2B + HA + DP + GIDP, family = binomial,
## data = bbproj_trn)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.31485 -0.25596 -0.04893 -0.00565 2.90172
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -37.894889 6.186389 -6.126 9.04e-10 ***
## W 0.392678 0.048963 8.020 1.06e-15 ***
## X2B -0.024866 0.007752 -3.208 0.00134 **
## HA 0.006988 0.002855 2.447 0.01440 *
## DP -0.024295 0.012063 -2.014 0.04401 *
## GIDP 0.021012 0.012242 1.716 0.08609 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 420.34 on 419 degrees of freedom
## Residual deviance: 182.92 on 414 degrees of freedom
## AIC: 194.92
##
## Number of Fisher Scoring iterations: 7step_aic_diag## $confusion_matrix
## actual
## prediction N Y
## N 70 8
## Y 2 10
##
## $sensitivity
## [1] 0.5555556
##
## $specificity
## [1] 0.9722222
##
## $misclassification
## [1] 0.1111111Optimal Cutoff
opt_step_aic <- opt_logistic_cutoff(step_aic_model,
cut_start = 0.1,
cut_end = 0.8,
plotit = FALSE)
opt_step_aic_diag <- classify_and_diagnose(step_aic_model,
cutoff = opt_step_aic["cutoff"])opt_logistic_cutoff(step_aic_model, cut_start = 0.1, cut_end = 0.8)
## cutoff misclass sensitivity specificity
## 0.23000000 0.07777778 0.94444444 0.91666667opt_step_aic_diag$confusion_matrix## actual
## prediction N Y
## N 66 1
## Y 6 17